The only algebraic integers which are found in the set of rational numbers are the integers. In other words, the intersection of Q{\displaystyle \mathbb {Q} } and A is exactly Z{\displaystyle \mathbb {Z} }. The rational number a/b is not an algebraic integer unless b divides a. Note that the leading coefficient of the polynomial bx − a is the integer b. As another special case, the square root √n of a non-negative integer n is an algebraic integer, and so is irrational unless n is a perfect square.

If d is a square free integer then the extension K=Q(d){\displaystyle K=\mathbb {Q} ({\sqrt {d}})} is a quadratic field of rational numbers. The ring of algebraic integers OK contains √d since this is a root of the monic polynomial x2 − d. Moreover, if d ≡ 1 (mod 4) the element (1 + √d)/2 is also an algebraic integer. It satisfies the polynomial x2 − x + (1 − d)/4 where the constant term(1 − d)/4 is an integer. The full ring of integers is generated by √d or (1 + √d)/2 respectively. See quadratic integers for more.

The ring of integers of the field F=Q[α],α=m3{\displaystyle F=\mathbb {Q} [\alpha ],\alpha ={\sqrt[{3}]{m}}} has the following integral basis, writing m=hk2{\displaystyle m=hk^{2}} for two square-free coprime integers h and k:[1]

If P(x) is a primitive polynomial which has integer coefficients but is not monic, and P is irreducible over Q{\displaystyle \mathbb {Q} }, then none of the roots of P are algebraic integers. (Here primitive is used in the sense that the highest common factor of the set of coefficients of P is 1; this is weaker than requiring the coefficients to be pairwise relatively prime.)

The sum, difference and product of two algebraic integers is an algebraic integer. In general their quotient is not. The monic polynomial involved is generally of higher degree than those of the original algebraic integers, and can be found by taking resultants and factoring. For example, if x2 − x − 1 = 0, y3 − y − 1 = 0 and z = xy, then eliminating x and y from z − xy and the polynomials satisfied by x and y using the resultant gives z6 − 3z4 − 4z3 + z2 + z − 1, which is irreducible, and is the monic polynomial satisfied by the product. (To see that the xy is a root of the x-resultant of z − xy and x2 − x − 1, one might use the fact that the resultant is contained in the ideal generated by its two input polynomials.)

Any number constructible out of the integers with roots, addition, and multiplication is therefore an algebraic integer; but not all algebraic integers are so constructible: in a naïve sense, most roots of irreducible quintics are not. This is the Abel-Ruffini theorem.

Every root of a monic polynomial whose coefficients are algebraic integers is itself an algebraic integer. In other words, the algebraic integers form a ring which is integrally closed in any of its extensions.